Question: Can Langlands Reciprocity be used to prove Fermat's Last Theorem?
A few years ago I was reading a book on the Langlands Program and the introduction provided a list of motivating consequences. Among the usual suspects was the claim that it would provide a simple proof to Fermat's Last Theorem.
At the time, this tidbit had little to do with my purpose for reading the book; I stored it as a curiosity and moved on. At some point I casually brought it up to a researcher. The response was something along the lines of "Sure. Just apply reciprocity and it falls out."
I am curious again, so I have attempted to find some explanation. Unfortunately, Wiles' proof is often designated as part of the Langlands Program itself, so searching the relevant terms yields expository resources directed at that work. My question is directed at alternative approaches to FLT which have been forgotten in the wake of Wiles' success.
Unfortunately, I do not remember the book which provoked the question.